Why did the Old Man kill his own father?

The Old Man killed his father because the latter squandered the wealth that belonged to the Old Man's mother. In the Old Man's mind, his father (by squandering his wife's wealth) also defrauded his child (the Old Man) of his rightful inheritance.
In the play, the Old Man tells his sixteen-year-old son that his mother once owned a grand mansion. In her youth, she had fallen in love with one of her grooms. She married him, despite her own mother's wishes. After the marriage, her mother disowned her and never spoke to her again.
The Old Man relates that his mother never knew how his father squandered her wealth. She died while giving birth to the Old Man. After her death, the Old Man's father spent the wealth of her estate on horses, women, drink, and cards. To pay for his accumulating debts, the Old Man's father even had the trees on the property cut down. The Old Man tells his son that he will never forgive his father for laying waste to an illustrious mansion, especially one that was inhabited by the great men of his age. The Old Man describes his father's actions as a "capital offence."
As we can see, the Old Man still resents his father for squandering his mother's fortune, and he also hates him for defrauding him of his inheritance. The Old Man is also angry that his father never provided him with an education. Instead, the Old Man had to learn how to read from a gamekeeper's wife, and he had to rely on a Catholic curate in order to learn Latin. Eventually, the Old Man had to earn a living as a lowly peddler. So, the Old Man killed his father for what he considers were immoral actions against him and his mother.

The Old Man's father was a dissolute wastrel, a chronic alcoholic who shamelessly frittered away the family fortune. As a boy, the Old Man was prevented by his father from attending school, and so to this day, he still harbors resentment towards him for his ignorance. The Old Man's son seems unconcerned by his late grandfather's character. Rather, he appears more interested in the glittering world of wealth and privilege his family once inhabited.
At the same time, the young man is aware of certain unpleasant rumors concerning his father, the Old Man. One night, the Old Man's father got drunk as usual, only this time he burned down the family home. The boy has heard that the Old Man killed his father that night. When asked about it, the Old Man admits that the rumors are indeed true.
It would appear that the Old Man was motivated primarily by revenge. His mother married beneath herself and the result was the destruction of a great family and its ancestral home. The Old Man is also eaten up with resentment over his reduced status in life, directly caused by his father's profligacy. Although he couldn't have foreseen how his life would turn out when he stabbed his father to death amid the burning wreckage of the family home, the years since have confirmed to the Old Man's satisfaction that he made the right decision.